The Davenport constant of balls and boxes

2025/10/23 by Benjamin Girard, Girard, Benjamin, Alain Plagne +1
Mathematics · #Limits and Structures in Graph Theory #Point processes and geometric inequalities #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2510.20412

Abstract

Given an additively written abelian group G and a set X⊆ G, we let D(X) denote the Davenport constant of X, namely the largest non-negative integer n for which there exists a sequence x₁, …, x_n of elements of X such that ∑_i=1ⁿ x_i =0 and ∑_{i ∈ I} x_i ≠ 0 for each non-empty proper subset I of \1, …, n\. In this paper, we mainly investigate the case when G is ℤ² and ℤ³, and X is a discrete Euclidean ball. An application to the classical problem of estimating the Davenport constant of a box - a product of intervals of integers - is then obtained.

Citations

• Monoids of modules and arithmetic of direct-sum decompositions
• A zero-sum theorem over Z
• On the arithmetic of Krull monoids with infinite cyclic class group
• The Difference Between Consecutive Primes, II

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